3.577 \(\int \frac{a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=24 \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]

[Out]

x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]

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Rubi [A]  time = 0.0226305, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022 \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + 2*(b*c + a*d)*x^2 + 3*b*d*x^4)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]

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Rubi in Sympy [A]  time = 31.2192, size = 20, normalized size = 0.83 \[ x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*c+2*(a*d+b*c)*x**2+3*b*d*x**4)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

x*sqrt(a + b*x**2)*sqrt(c + d*x**2)

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Mathematica [A]  time = 0.076475, size = 24, normalized size = 1. \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + 2*(b*c + a*d)*x^2 + 3*b*d*x^4)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]

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Maple [A]  time = 0.013, size = 21, normalized size = 0.9 \[ x\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*c+2*(a*d+b*c)*x^2+3*b*d*x^4)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)

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Maxima [A]  time = 1.54164, size = 27, normalized size = 1.12 \[ \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*b*d*x^4 + 2*(b*c + a*d)*x^2 + a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="maxima")

[Out]

sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x

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Fricas [A]  time = 0.218286, size = 27, normalized size = 1.12 \[ \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*b*d*x^4 + 2*(b*c + a*d)*x^2 + a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="fricas")

[Out]

sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a c + 2 a d x^{2} + 2 b c x^{2} + 3 b d x^{4}}{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*c+2*(a*d+b*c)*x**2+3*b*d*x**4)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral((a*c + 2*a*d*x**2 + 2*b*c*x**2 + 3*b*d*x**4)/(sqrt(a + b*x**2)*sqrt(c +
d*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 \, b d x^{4} + 2 \,{\left (b c + a d\right )} x^{2} + a c}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*b*d*x^4 + 2*(b*c + a*d)*x^2 + a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)),x, algorithm="giac")

[Out]

integrate((3*b*d*x^4 + 2*(b*c + a*d)*x^2 + a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)
), x)